Machine Design Databook Episode 1 part 4 pptx

c distance from neutral axis to extreme ﬁbre, mvelocity of propagation of plane wave along a thin bar, m/s cL velocity of propagation of plane longitudinal waves in an force acting on pi

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TABLE 2-10

Mechanical and physical constants of some materials1;2

Modulus of elasticity, E Modulus ofrigidity, G

b ¼ weight density; w is also the symbol used for unit weight of materials.

Sources: K Lingaiah and B R Narayana Iyengar, Machine Design Data Handbook, Vol I (SI and Customary Metric Units), Suma Publishers, Bangalore, India, and K Lingaiah, Machine Design Data Handbook, Vol II (SI and Customary Metric Units), Suma Publishers, Bangalore, India 1986.

STATIC STRESSES IN MACHINE ELEMENTS 2.39

STATIC STRESSES IN MACHINE ELEMENTS

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¼ G r L

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1 Maleev, V L and J B Hartman, Machine Design, International Textbook Company, Scranton,Pennsylvania, 1954

2 Shigley, J E., Mechanical Engineering Design, 3rd edition, McGraw-Hill Book Company, New York, 1977

3 Lingaiah, K., and B R Narayana Iyengar, Machine Design Data Handbook, Vol 1 (SI and Customary MetricUnits), Suma Publishers, Bangalore, India, 1986

4 Lingaiah, K., Machine Design Data Handbook, Vol II (SI and Customary Metric Units), Suma Publishers,Bangalore, India, 1986

5 Lingaiah, K., Machine Design Data Handbook, McGraw-Hill Publishing Company, New York, 1994

6 Ashton, J E, J C Halpin and P H Petit, Primer on Composite Materials-Analysis, Technomic PublishingCo., Inc., 750 Summer St., Stanford, Conn 06901, 1969

7 Roark, R J., and W C Young, Formulas for Stress and Strain, McGraw-Hill Publishing Company, NewYork, 1975

8 Hertz, H., On the Contact of Elastic Solids, J Math (Crelle’s J.) Vol 92, pp 156–171, 1981

9 Hertz, H., On Gesammelte werke, Vol I., p 155, Leipzig, 1895

10 Timoshenko, S., and J N Goodier, Theory of Elasticity, McGraw-Hill Book Company, New York, 1951

BIBLIOGRAPHY

1 Black, P H., and O Eugene Adams, Jr., Machine Design, McGraw-Hill Book Company, New York, 1965

2 Lingaiah, K, and B R Narayana Iyengar, Machine Design Data Handbook (fps units), Engineering College Operative Society, Bangalore, India, 1962

Co-3 Norman, C A., E S Ault, and I F Zarobsky, Fundamentals of Machine Design, The Macmillan Company,New York, 1951

4 Vallance, A E., and V L Doughtie, Design of Machine Members, McGraw-Hill Book Company, New York,1951

5 Timosheko, S., and J N Goodier, Theory of Elasticity, McGraw-Hill Book Company, New York, 1951

6 Timoshenko, S., and J M Gere, Mechanics of Materials, Van Nostrand Reinhold Company, New York, 1972

7 George Lubin, Editor, Handbook of Composites, Van Nostrand Reinhold Company, New York, 1982

8 John Murphy, Reinforced Plastic Handbook, 2nd edition, Elsevier, Advanced Technology, 1998

9 Hamcox, N L., and R M Mayer, Design Data for Reinforced Plastics, Chapman and Hall, 1994

STATIC STRESSES IN MACHINE ELEMENTS 2.49

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c distance from neutral axis to extreme ﬁbre, m

velocity of propagation of plane wave along a thin bar, m/s

cL velocity of propagation of plane longitudinal waves in an

force acting on piston due to steam or gas pressure corrected for

inertia eﬀects of the piston and other reciprocating parts, kN

F centrifugal force per unit volume, kN/m3

Fc the component of F acting along the axis of connecting rod, kN

Fi inertia force, kN

Fic inertia force due to connecting rod, kN

Fir inertia force due to reciprocating parts of piston, kN

Fs static load, kN

g acceleration due to gravity, 9.8066 m/s2

height of fall of weight, m

J polar moment of inertia, m4(cm4)

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radius of curvature of the path of motion of mass, m

the moment arm of the load, m

internal elastic energy, N m

Ui work done in case of suddenly applied load, N m

Umax maximum internal elastic energy, N m

Up potential energy, N m

V velocity of particle in the stressed zone of the bar, m/s

volume, m3

V0 initial velocity at the time of impact, m/s

w speciﬁc weight of material, kN/m3

Z section modulus, m3(cm3)

angle between the crank and the centre line of connecting rod, deg

unit shear strain, rad/rad

weight density, kN/m3

deﬂection/deformation, m (mm)

i deformation/deﬂection under impact action, m (mm)

s static deformation/deﬂection under the action of weight, m (mm)

" unit strain also with subscripts,mm/m

"x,"y,"z strains in x, y, and z-directions,mm/m

xy,yz,zx shearing-strains in rectangular coordinates, rad/rad

angle between the crank and the centre line of the cylinder

measured from the head-end dead-centre position, degstatic angular deﬂection, deg

angle of twist, deg

i angular deﬂection under impact load, deg

, Lame´’s constants

normal stress (also with subscripts), MPa

i impact stress (also with subscripts), MPa

0 initial stress at the time of impact and velocity V0, MPa

x,y,z normal stress components parallel to x, y, and z-axis

shearing stress, MPa

l time of load application, s

n period of natural frequency, s

xy, yz, zx shearing stress components in rectangular coordinates, MPa

! angular velocity, rad/s

Note: s and swith ﬁrst subscript s designate strength properties of material used in the design which will be used and followed throughout the book Other factors in performance or in special aspects which are included from time to time in this book and being applicable only in their immediate context are not given at this stage.

3.2 CHAPTER THREE

DYNAMIC STRESSES IN MACHINE ELEMENTS

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The internal elastic energy or work done when a

machine member is subjected to a gradually applied

load, Fig 3.1

The work done in case of suddenly applied load on an

elastic machine member (Fig 3-2)

FIGURE 3-1 Plot of force against deﬂection in case of

elas-tic machine member subject to gradually applied load.

The relation between suddenly applied load and

gra-dually applied load on an elastic machine member to

produce the same magnitude of deﬂection

DYNAMIC STRESSES IN MACHINE ELEMENTS 3.3

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The static deformation or deﬂection

IMPACT STRESSES

Impact from direct load

Kinetic energy

Impact energy of a body falling from a height h

The height of fall of a body that would develop the

velocity v

The maximum stresses produced due to fall of weight

Wthrough the height h from rest without taking into

account the weight of shaft and collar (Fig 3-3)

FIGURE 3-3 Striking impact of an elastic machine

member by a body of weight W falling through a height h.

The maximum deﬂection or deformation of shaft due

to fall of weight W through the height h from rest

neglecting the weight of shaft and collar

The stress produced due to suddenly applied load

The maximum deﬂection or deformation produced by

suddenly applied load

h ¼v2

s24

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The kinetic energy taking into account the weight of

shaft or bar and collar

The relation between, , F and W

The maximum stress due to fall of weight W through

the height h from rest taking into account the weight

of shaft/bar and collar

The maximum deﬂection due to fall of weight W

through the height h from rest taking into

considera-tion the weight of shaft/bar and collar

Internal elastic energy of weight W whose velocity v is

3W

ð3-15Þwhere Vc¼ velocity of collar and weight W afterthe load striking the collar, m/s

where Wb¼ weight of shaft or barF

s24

1

1þ ðWb=3WÞ

s24

35

s24

max¼ WL

AE 1þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ2hEAWL

1

1þ ðWb=3WÞ

s

24

35

s24

3

U ¼Wv2

whereplane, deg

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FIGURE 3-4 Impact by a falling body

The equation for energy balance for an impact by a

falling body (Fig 3-4)

Another form of equation for deformation or

deﬂec-tion in terms of velocity v at impact

Equivalent static force that would produce the same

maximum values of deformation or deﬂection due

to impact

BENDING STRESS IN BEAMS DUE TO

IMPACT

Impact stress due to bending

FIGURE 3-5 Impact by a falling body on a cantilever beam

Deﬂection of the end of cantilever beam under impact

(Fig 3-5)

The maximum bending stress for a cantilever beam

taking into account the total weight of beam

s0

@

1A

s0

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The maximum deﬂection at the end of a cantilever

beam due to fall of weight W through the height h

from rest taking into consideration the weight of beam

The maximum bending stress for a simply supported

beam due to fall of a load/weight W from a height h

at the midspan of the beam taking into account the

total weight of the beam (Fig 3-6)

FIGURE 3-6 Simply supported beam

The maximum deﬂection for a simply supported beam

due to fall of a weight W from a height h at the

mid-span of the beam taking into account the weight of

beam (Fig 3-6)

TORSION OF BEAM/BAR DUE TO IMPACT

(Fig 3-7)

The equation for maximum shear stress in the bar due

to impact load at a radius r of a falling weight W from

a height h neglecting the weight of bar

FIGURE 3-7 Twist of a beam/bar

max¼ st 1þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ2hst

s24

1

1þ ð17 =35Þ

s

24

35

max¼ st 1þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ2hst

s24

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The equation for angular deﬂection or angular twist

of bar due to impact load W at radius r and falling

through a height h neglecting the weight of bar

LONGITUDINAL STRESS-WAVE IN

ELASTIC MEDIA (Fig 3-8)

One-dimensional stress-wave equation in elastic

media (Fig 3-8)

For velocity of propagation of longitudinal

stress-wave in elastic media

The solution of stress-wave Eq (3-33a)

The value of circular frequency p

The frequency

LONGITUDINAL IMPACT ON A LONG

BAR

The velocity of particle in the compression zone

The uniform initial compressive stress on the free end

s24

3

@2u

s

¼

ffiffiffiffiE

p ¼nc

l ¼nl

ffiffiffiffiffiffiEg

s

¼nl

ffiffiffiffiE

s

ð3-35aÞwhere n is an integer ¼ 1; 2; 3;

f ¼ p

2¼

n2l

ffiffiffiffiE

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The equations of motion in terms of three

displace-ment components assuming that there are no body

forces

FIGURE 3-9 Prismatic bar subject to suddenly applied

uniform compressive stress

Dilatational and distortional waves in

isotropic elastic media

From the classical theory of elasticity equations for

irrotational or dilatational waves

Equations for distortional waves

Equations (3-40) to (3-41) are one-dimensional stress

wave equations of the form

The velocity of stress wave propagation for the case of

@t2¼ þ 2G r2

@2v

@t2 ¼ þ 2G r2

@2u

@t2¼G r2

@2v

@t2¼G r2

@2w

s

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEð1 Þð1 þ Þð1 2Þ

s

ð3-43Þ

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The velocity of stress wave propagation for the case of

zero volume change

The ratio of c1to c2

The velocity of stress wave propagation for a

trans-verse stress wave, i.e distortional wave in an inﬁnite

plate

The velocity of stress wave propagation for plane

longitudinal stress wave in case of an inﬁnite plate

TORSIONAL IMPACT ON A BAR

Equation of motion for torsional impact on a bar

FIGURE 3-10 Torsional impact on a uniform bar showing

torque on two faces of an element

The angular velocity of the end of a bar subject to

tor-sion relative to the unstressed region

The shear stress from Eq (3-50)

The initial shear stress, if the rotating body strikes the

end of the bar with an angular velocity!0

a ¼ c2¼

ffiffiffiffiG

s

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE2ð1 Þ

s

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE

s

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE ð1 2Þ

s

ð3-47Þ

@2

@t2¼ c2 t

@2

ct¼

ffiffiffiffiffiffiGg

s

¼

ffiffiffiffiG

t

d ffiffiffiffiffiffi G

!d2

ffiffiffiffiffiffi G

p

ð3-51aÞ

0¼!0d2

ffiffiffiffiffiffi G

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The maximum shear stress for the case of a shaft ﬁxed

or attached to a very large mass/weight at one end and

suddenly applied rotational load at the other end by

means of some mechanical device such as a jaw

clutch (Fig 3-11)

FIGURE 3-12 A striking rotating weight with

mass-moment of inertia I rotating at ! 0 engages with one end of

shaft and the other end of shaft ﬁxed to a mass-moment of

inertia I f

The more accurate equation for the max which is

based on stress wave propagation

The initial/maximum ( i max) shear stress for the

case of a system shown in Fig 3-12

A similar equation to Eq (3-54) for maximum stress

for longitudinal impact

Accurate maximum stress for longitudinal impact

stress based on stress wave propagation as suggested

by Prof Burr

Accurate maximum stress for torsional impact shear

stress based on stress-wave propagation as suggested

by Prof Burr

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

1

1þ 3

0

@

1A

vut

26

I ¼ mass moment of inertia of striking rotating weight

Iband I correspond to Wband W of the weight of thebar and the rotating mass or weight respectively

max 0 1þ

ffiffiffiffiffiffiffiffiffiffiffi1 þ

23

s24

3

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INERTIA IN COLLISION OF ELASTIC

BODIES

When a body having weight W strikes another body

that has a weight W0, impact energy Wh is reduced

to nWh, according to law of collision of two perfectly

inelastic bodies, the formula for the value of n

RESILIENCE

The expression for resilience in compression or

tension

The modulus of resilience

The area under the stress-strain curve up to yielding

point represents the modulus of resilience (Fig 1.1)

The resilience in bending

The modulus of resilience in bending

Resilience in direct shear

The modulus of resilience in direct shear

Resilience in torsion

The modulus of resilience in torsion

The equation for strain energy due to shear in bending

The modulus of resilience due to shear in bending

V

E¼12

2AL

2

2 b

whereðk=cÞ2¼ 1 for rectangular cross-section

¼ 1 for circular section

c ¼ distance from extreme fibre toneutral axis

k0c

2

ð3-67Þ

ðl0

k F2

u ¼k

2 e

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The equation for shear or distortional strain energy

per unit volume associated with distortion, without

change in volume

The equation for dilatational or volumetric strain

energy per unit volume without distortion, only a

change in volume

For maximum resilience per unit volume (i.e., for

modulus of resilience), resilience in tension for

var-ious engineering materials and coeﬃcients a and b;

velocity of propagation c and ct

Modulus of rigidity, G c ¼

ffiffiffiffi E

s

¼

ffiffiffiffiffiffi Eg

c t ¼

ffiffiffiffi G

s

¼

ffiffiffiffiffiffi Gg

Material g/cm3 lb m /in3 kN/m3 GPa Mpsi GPa Mpsi m/s ft/s m/s ft/s Aluminum alloy 2.71 0.098 26.6 71.0 10.3 26.2 3.8 5116 16785 3110 10466

Carbon steel 7.81 0.282 76.6 206.8 30.0 79.3 11.5 5145 16887 3200 10485 Cast iron, gray 7.20 0.260 70.6 100.0 14.5 41.4 6.0 3727 12223 2407 7865

#Note: ¼ Mass density, g/cm 3

(lb m /in3), ¼ weight density (speciﬁc weight), kN/m 3

(lbf/in3), g ¼ 9:8066 m/s2in SI units, g ¼ 980 in=s2¼ 32:2 ft=s 2

in fps units.

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TABLE 3-2

Maximum resilience per unit volume (2, 1)

e

2E Shear, simple transverse

2 e

2G Bending in beams

With simply supported ends:

Concentrated center load and rectangular cross-section 2

e

18E Concentrated center load and circular cross-section 2

e

6E Fixed at both ends:

Solid round bar

2 e

4G Hollow round bar with D 0 greater than D i

2 e

3.14 CHAPTER THREE

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TABLE 3-3

Coeﬃcients in Eq (3-58) (1)

Center impact on single beam 17 5

Center impact on beam with ﬁxed ends 13 1

End impact on cantilever beam 4

Cast iron has no well-deﬁned elastic limit, but the values may be safely used anyway for all practical purposes.

Source: Reproduced courtesy of V L Maleev and J B Hartman, Machine Design, International Textbook Co., Scranton, Pennsylvania, 1954.

Carbon steel 7. 81 0.282 76.6 206.8 30.0 79.3 11 .5 5 14 5 16 887 3200 10 48 5 Cast iron, gray 7.20 0.260 70.6 10 0.0 14 .5... 0 1? ??

1 ỵ

23

s 24

3

DYNAMIC STRESSES IN MACHINE ELEMENTS 3 .11

DYNAMIC STRESSES IN MACHINE. .. Hartman, Machine Design, International Textbook Co., Scranton, Pennsylvania, 19 54.

DYNAMIC STRESSES IN MACHINE ELEMENTS 3 .15

DYNAMIC STRESSES IN MACHINE

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